OPTIMAL CONDITIONS FOR CAMERA CALIBRATION USING A PLANAR TEMPLATE

Carlos Ricolfe-Viala, Antonio-Jose Sanchez-Salmeron

Instituto de Automática e Informática Industrial

Universitat Politècnica de València, Spain

{cricolfe,asanchez}@isa.upv.es; http://robotica.ai2.upv.es/

ABSTRACT

Different techniques exist for camera calibration based on

photogrammetry or self-calibration. Much work has been

done to obtain intrinsic and extrinsic camera parameters and

also in data pre- and post-processing techniques. From a

practical viewpoint, it is quite difficult to decide which

calibration method produces the best results and even

whether any data processing is necessary.

This paper proposes a set of optimal performance

conditions to resolve the calibration process accurately. The

calibration method based on a 2D pattern has been chosen

since camera is calibrated using several images of a planar

template easily. The proposed optimal conditions define the

number of points to design the calibration pattern, the

number of images to resolve the calibration accurately, as

well as positions and orientations from where images should

be taken. The recommend rules will help to obtain a well-

conditioned calibration process.

Index Terms— Camera calibration, 2D pattern, non-

linear calibration, optimal conditions

1. INTRODUCTION

In the calibration process, it is essential to consider the

quality of results, which obviously depends on data

accuracy. Some studies have been made on calibration errors

committed due to imprecise measurement of the image plane

or calibration template [1, 2, 3, 4]. The calibration process is

also affected by erroneous association of one point in reality

with a point in the image. Some authors use statistical tools

to detect these anomalies [5]. Computing errors due to

instabilities of the mathematical tool should also be taken

into account [6]. In these cases, data normalization improves

the robustness of the algorithm and gives more accurate

results [7]. Another significant case is the computed camera

model. A very complex model can complicate the algorithm

and the result obtained will not be much better than that

obtained with a simpler model. A very complicated model

may produce instabilities in the process of questing and

produce absurdities [8].

The state of the art of calibration provides some guide

about efficiency of camera calibration in all situations. The

Tsai method [9] represents a classical calibration process

based on the measurements of the 3D points in the template

taking a fixed reference. This method has been widely used

in the past. Salvi [10] compares the calibration methods

developed between 1982 and 1998, with the Tsai method

showing better performance, despite the fact that it requires

high precision in input data. On the other hand, Zhang’s

method [3], which is not included in Salvi’s comparison

[10], represents a new era in the camera calibration process.

This method uses images of a 2D template taken from

different camera positions and orientations. In this way, the

advantages of camera self-calibration are combined with the

point coordinate-based calibration. This calibration method

is highly flexible, since the camera and the template can be

moved freely and also as many images as are required can be

taken without measuring any position of the template. Sun

[11] compares the Tsai method with Zhang’s method. On

one hand, Tsai produces a precise estimation of camera

parameters if the input data have not been corrupted. Since

100 points in the template are necessary and the coordinates

should be referred to a fixed origin, careful design of the

calibration template and a very accurate coordinate

measurement are necessary. Nevertheless, errors are too

easily committed and in practice these results are not as

accurate as expected, as shown by Sun [11]. On the other

hand, Zhang’s method based on a 2D template requires

neither a special design nor precise point measurement. Sun

obtains camera calibration with a hand-made template, and

better results are computed using Zhang’s method.

Furthermore, the sensibility of the calibration algorithm to

errors in the measures can be improved by increasing the

spotted number in the template, by simply printing a

chessboard with more corners. The results of the comparison

show the flexibility and adaptability of Zhang’s method, as it

can be performed on any scene. Considering the results of

these two authors, Zhang’s method is used as a reference for

camera calibration.

This paper proposes the optimum conditions from the

viewpoint of number of points, number of images and

location of the camera for taking the images, to improve on

Zhang’s camera calibration method. These definitions obtain

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a well-conditioned calibration process which compute better

results. The paper is structured as follows: Section 2 defines

optimal conditions for camera calibration from the

viewpoint of number of points, number of images and

location of the camera taking the images. In section 3

experimental results are presented. Paper ends with some

conclusions.

2. OPTIMAL CONDITIONS FOR CAMERA

CALIBRATION

To calibrate the linear camera model, Zhang [3] describes a

method based on the homographies between a planar

calibration pattern and its images from several camera

positions. For each homography two homogeneous

equations arise as:

012

2211 =

−b

v

vvT

TT

(1)

where: [

]3332233113

22122111

jijijijiji

jijijijiT

ij hhhhhhhhhh

hhhhhhhhv

++

+=

(2)

h represents elements of the homography H and b=[b11, b12,

b22, b13, b23, b33], where bij represents the element ij of the

symmetric matrix K-T

·K-1

. K contains intrinsic parameters of

pin-hole model. If m images of the calibration object are

observed, m equations such as (1) arise, giving V·b=0, where

V is a 2mx6 matrix. At least three images are necessary m≥3

in order to obtain a unique solution. The closed-form

solution is given by the eigenvector of VTV associated with

the smallest eigenvalue. Once b is estimated, camera internal

parameters can be computed. When K is known, external

parameters for each image can be computed when the

corresponding homography is known. See [3] for details.

2.1. Camera calibration scene

To define the optimal positions for image capture, a

calibration scene is defined. (o,{xw,yw,zw}) is 3D scene

coordinate system and (o’,{xc,yc,zc}) is camera coordinate

system located as shown in figure 1. The centre of the

template is situated in the origin of coordinates of the scene

and the camera is always translated with a negative z-

coordinate. The orientation of the camera is defined using

the position. This means that the camera optical axis crosses

the centre of the template always as shown on figure 1.

Moreover, Xc axis of the camera coordinate system is always

parallel to plane Xw-Yw of the scene. From a practical point

of view, the template is located on the floor of the

calibration scene and the camera is located on a tripod,

which keeps the upper border of the image parallel to the

floor. The origin of the scene coordinate system is the

template centre.

Figure 1. Scene and camera coordinate systems. Orientation is

computed starting with camera location. Optical axis goes through

origin of coordinates in the scene.

To relate orientation with position of the camera, two

rotations are defined. First camera and scene coordinate

systems coincide. Then, the camera is rotated at angle β with

respect to the Zc axis as shown in figure 1. Second, in

relation to the Xc axis, the camera is rotated an angle α as

shown on figure 1. These two rotations keep the Xc axis

parallel to plane X-Y of the calibration scene. These angles

are a function of the camera position:

1

cosm

t y=β

1

sinm

tx=β

2

cosm

tz=α

2

1sinm

m=α

22

1 yx ttm += 222

2 zyx tttm ++=

(3)

where m1 represents the module of camera translation in

plane Xw-Yw, symbolizing the floor, and m2 is the distance of

the camera to the origin of coordinates in the scene. The

origin of coordinates of the image is in the centre and will

therefore be pixels with negative coordinates. Another

supposition refers to the scale factor ku and kv in image axes.

It is assumed that pixels are squared and therefore the scale

factor is the same in both axes and we rename with k.

2.2. Condition of camera calibration

As known in a general framework, a matrix M has r eigen

vectors and r eigen values, where r is the rank of matrix M.

Eigen vectors are orthogonal to one another. Eigen vectors

modules represent the number of vectors in the matrix M

oriented in the direction defined by each eigen vector. The

module of one eigen vector increases if there are more

vectors in matrix M oriented in this eigen vector direction.

The ratio between the bigger module and the smaller module

is called the condition number of the matrix. To obtain a

well-conditioned system, the condition number should be

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one. This means there are vectors in M which cover all R-

dimensional space. Therefore, if a parameter vector is

estimated with this matrix M, information from the data

affects all dimensions of the parameter vector. In the event

of a badly conditioned matrix, some parameters are not

influenced by the information contained in matrix M and

therefore such estimation will be erroneous.

If this theory is particularized to linear camera

calibration, the camera parameters are in the eigen vector

associated with the smaller eigen value of matrix V defined

by expression (1). Matrix V is formed from the elements of

homographies lH. Since the condition number of matrix V

should be one, all vectors of matrix V should have the same

module, and they should be orthogonal to one another.

Vectors of V depend on the homographies lH, whereas

lH

depends on the locations from which images are taken.

Therefore, vectors of V depend indirectly on positions in

which the camera is located. Consequently, we can say that

positions from which images are taken, influence the

condition of matrix V. As a result, to obtain a well-

conditioned matrix V several locations for the camera should

be defined.

Dimension of V is 2mx6. Therefore the minimum

number m of homographies to resolve the system is m≥3.

Within this framework, a well-conditioned matrix V will

arise with vectors lvij, which are orthogonal to one another,

with equal modules. Sub index ij represents vector v formed

from the columns i-th and j-th of homography l. Thus,

homographies forming a set of vectors lvij orthogonal to one

another and with identical modules are required. In

consequence care must be taken when dealing with

homography elements.

To obtain a well-conditioned matrix V, the following

expressions must be true for all i, j, l.

( ) 0· 122211 =− vvv lTlTl

(4)

( )( ) 0·· 121222112211 =−−− vvvvvv lTlllTlTl

(5)

The first expression represents an orthogonal condition

and the second stands for module equality. i, j, have values 1

and 2 since they correspond with the two left-hand columns

of the homography l. The number of homographies l should

be at least 3. If 3 homographies are established whose

vectors are orthogonal and with equal modules, a well-

conditioned matrix V will be obtained with only a minimum

amount of information.

2.3. Locations for image capturing

Analysing expressions (4) and (5) they are satisfied when a

camera is located with null coordinates ltx or

lty. If

ltx is zero,

lty and ltz will not be zero. Also, if lty is zero, ltx and ltz will

not be zero. ltx or

lty are set different to zero moving the

camera along the X or Y scene axis. The camera position in

the Z axis of the scene ltz is computed with the following

expressions depending on whether the camera has been

located with coordinate ltx or

lty different to zero:

( )( )4121

4121412121

1

·31

·12·43·2·21·2

xx

xxxxx

ztt

tttttt

++

+−−+−−−−±=

( )( )

4121

4121412121

1

·31

·12·43·2·21·2

yy

yyyyy

ztt

tttttt

++

+−−+−−−−±=

(6)

From a practical point of view, the camera is moved

along one of the axes representing the X or Y axis of the

coordinate system of the scene and then the altitude of the

camera in the Z axis is defined using the expression (6).

After this, the camera must be oriented bearing in mind that

the optical axis passes through the origin of coordinates in

the scene. The camera must be moved along the X or Y axis

exactly the same distance. It is important to note that

expressions (6) do not depend on intrinsic parameters. This

means that such camera locations do not depend on camera

features and can be used with any camera.

3. EXPERIMENTAL RESULTS

To test the performance of the camera calibration conditions

a simulated camera is similar to Zhang in [3] with

parameters ku=1250, kv=900, u0=250, v0=250 and θ=1.09083

equivalent to 89.95º is used. Image resolution is 512x512

pixels. The calibration template is a chessboard of

10x14=140 corners. Template size is 180x250 mm. In this

case, it is situated on the floor and all corners have

coordinates zw=0. The camera is oriented so as to view the

origin of coordinates in the centre of the image. Also,

camera orientation is computed to ensure the x-coordinate

axis of the image is parallel to the template plane. This

simulates the effect of taking an image of a template situated

on the floor using a tripod.

To calibrate the camera under optimal conditions, the

simulated camera is located at t1=(200, 0, 401.8), t2=(-100,

0, 200.9) and t3=(0, 300, 602.1). Also, camera is calibrated

using images from random positions t1=(150, 200, 580),

t2=(-50, 250, 880) and t3=(100, -20, 820). Both calibration

results changing the number of points in the template are

compared. Results are shown in figure 2 for intrinsic camera

parameters ku, u0 only. Similar performance has been

computed for the remaining camera parameters. Camera

calibration is solved using the linear and the non-linear

process. In all cases, computed camera parameters are

improved if optimal conditions are used.

The number of points necessary for the calibration

process can be defined analysing experimental results. If

images are taken taking into account optimal conditions,

more than 70 does not reduce parameters errors

significantly. Obviously, since constructing a calibration

template is an easy task, more than 70 points can be used to

improve the results.

2011 18th IEEE International Conference on Image Processing

855

Figure 2. Comparison of intrinsic camera parameters calibration

errors under optimal and non- optimal calibration process

conditions changing the number of points

4. CONCLUSIONS

Optimal conditions for camera calibration using a 2D pattern

have been defined. Camera parameters are computed the

eigen vector associated with the smaller eigen value of the

matrix composed with elements of several homographies.

The calibration process will therefore be more stable if the

condition number of this matrix is close to 1. To obtain this

condition number closer to 1, elements of the homographies

should be taken into account. Therefore, images of the

template should be captured from specific locations to

obtain a well-conditioned calibration process. To define the

optimal locations from where to take images of the template,

the condition number of this matrix has been analysed. The

camera is located taking into account that the altitude should

be twice its separation from the origin of coordinates in the

scene. Also, the camera should be located along the X or Y

axes of the scene coordinate system. The camera orientation

is defined assuming that the optical axis goes through the

calibration template. Finally, coordinates in the image and

the template should be referred to the centre of the image

and the template, respectively.

Although Zhang's method was intended for self-

calibration in which images are taken from anywhere, here

we propose a useful guide to improve the calibration results.

Acknowledgements

This work was partially funded by the Universidad

Politécnica de Valencia (PAID 2010-2431 and PAID

10017), Generalitat Valenciana (GV/2011/057) and by

Spanish government and the European Community under the

project DPI2010-20814-C02-02 (FEDER-CICYT) and

DPI2010-20286 (CICYT)

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